L(s) = 1 | − 2.38·3-s − 2.90·5-s − 2.69·7-s + 2.69·9-s + 1.35·11-s − 3.01·13-s + 6.93·15-s + 0.160·17-s − 19-s + 6.43·21-s − 4.12·23-s + 3.45·25-s + 0.737·27-s − 4.15·29-s − 3.19·31-s − 3.23·33-s + 7.84·35-s + 10.3·37-s + 7.19·39-s + 5.30·41-s + 3.84·43-s − 7.82·45-s + 8.59·47-s + 0.280·49-s − 0.383·51-s − 8.41·53-s − 3.93·55-s + ⋯ |
L(s) = 1 | − 1.37·3-s − 1.30·5-s − 1.01·7-s + 0.896·9-s + 0.408·11-s − 0.836·13-s + 1.79·15-s + 0.0389·17-s − 0.229·19-s + 1.40·21-s − 0.859·23-s + 0.690·25-s + 0.142·27-s − 0.771·29-s − 0.573·31-s − 0.562·33-s + 1.32·35-s + 1.70·37-s + 1.15·39-s + 0.828·41-s + 0.585·43-s − 1.16·45-s + 1.25·47-s + 0.0401·49-s − 0.0536·51-s − 1.15·53-s − 0.530·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 + 2.38T + 3T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 - 0.160T + 17T^{2} \) |
| 23 | \( 1 + 4.12T + 23T^{2} \) |
| 29 | \( 1 + 4.15T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 5.30T + 41T^{2} \) |
| 43 | \( 1 - 3.84T + 43T^{2} \) |
| 47 | \( 1 - 8.59T + 47T^{2} \) |
| 53 | \( 1 + 8.41T + 53T^{2} \) |
| 59 | \( 1 - 6.94T + 59T^{2} \) |
| 61 | \( 1 + 4.62T + 61T^{2} \) |
| 67 | \( 1 - 9.91T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 83 | \( 1 - 4.60T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 6.59T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.54524275633571495825493006072, −7.00077959594165574350610303270, −6.20508066901891181186176079468, −5.76545040120276570237035519582, −4.77267214743947520515310914957, −4.13788584618425938426795274195, −3.47578833399361901454444640649, −2.36083062462882708841531272475, −0.76585042365626124256548074126, 0,
0.76585042365626124256548074126, 2.36083062462882708841531272475, 3.47578833399361901454444640649, 4.13788584618425938426795274195, 4.77267214743947520515310914957, 5.76545040120276570237035519582, 6.20508066901891181186176079468, 7.00077959594165574350610303270, 7.54524275633571495825493006072